Gearbox fault diagnosis method based on physical information neural network

Abstract

The invention discloses a gearbox fault diagnosis method based on a physical information neural network. The method comprises the following steps: firstly, arranging a vibration signal acquisition system in a gear box, acquiring vibration and angular velocity signals, establishing an actual measurement data set with a label, and establishing a gear dynamic model to obtain a simulation data set with a label; then, training a physical information neural network formed by a long-short-term memory (LSTM) network by utilizing actual measurement and simulation data sets, and outputting a time-varying meshing stiffness curve by taking vibration and angular velocity signals as input; thirdly, extracting features from the meshing stiffness curve data to form a fault classification training set, and training a support vector machine (SVM) model by using the data set; and finally, inputting actually acquired vibration data into the physical information neural network and the SVM model to realize accurate diagnosis of the gearbox fault.

Description

Technical Field

[0001] The invention relates to the technical field of gearbox fault diagnosis, and in particular to a gearbox fault diagnosis method based on physical information neural network. Background Art

[0002] As the core component of the mechanical transmission system, the operating status of the gearbox is directly related to the reliability and production efficiency of the entire equipment. However, since the gearbox usually operates for a long time in a complex working environment and is affected by multiple factors such as load fluctuations, insufficient lubrication and component wear, the gearbox is very prone to failure. Common failures include gear wear, fatigue fracture, bearing damage, etc. Especially in large-scale industrial equipment, the failure of the gearbox will not only lead to production stagnation, but also may cause serious safety hazards. Therefore, how to diagnose gearbox failures in a timely and accurate manner is crucial to equipment maintenance and production safety.

[0003] Existing gearbox fault diagnosis methods mainly rely on a variety of sensor data such as vibration signals, temperature changes, and sound signals, and identify potential gearbox faults by analyzing the abnormal characteristics of these physical quantities. For example, vibration analysis can effectively monitor the vibration mode of the gears meshing inside the gearbox. When the gears are damaged, fatigued, or failed, their vibration signals will show obvious abnormalities. However, since vibration signals are easily interfered by external environmental noise, and different types of faults may show similar characteristics under complex working conditions, the accuracy of fault diagnosis is greatly limited.

[0004] In recent years, with the rapid development of deep learning technology, researchers have gradually applied neural networks to gearbox fault diagnosis. By building an end-to-end model to automatically extract features from vibration signals and perform classification and identification, the efficiency and accuracy of fault diagnosis have been greatly improved. Common neural network architectures include convolutional neural networks (CNN), recurrent neural networks (RNN), long short-term memory networks (LSTM), etc. However, these data-driven deep learning methods do not fully consider the physical characteristics of the gearbox system. The model lacks physical constraints during training, resulting in insufficient generalization ability, especially in complex working conditions and unseen fault modes. The diagnostic accuracy is still low.

[0005] To this end, a gearbox fault diagnosis method based on physical information neural network is proposed. By taking the dynamic equation of the gearbox as a physical constraint and embedding it into the training process of the neural network, the model not only relies on data learning, but also follows the physical laws of the gearbox system. This method can improve the generalization ability of the model, reduce the impact of noise interference on the diagnosis results, and realize accurate diagnosis of gearbox faults under complex working conditions. Summary of the invention

[0006] In view of this, the present invention aims to propose a gearbox fault diagnosis method based on physical information neural network, so as to realize efficient and accurate diagnosis of gearbox faults and improve the automation of its diagnosis. This method combines physical constraints with neural networks and effectively uses actual data and simulation data for modeling. It can accurately diagnose various gearbox faults, and the diagnostic results have good robustness and generalization ability.

[0007] The present invention is achieved through the following technical solutions:

[0008] A gearbox fault diagnosis method based on physical information neural network includes the following steps:

[0009] S1: Obtain the design parameters of the gears in the gearbox, arrange vibration signal sensors around the gearbox bearings, and arrange gear speed sensors on the top side of the gears;

[0010] S2: Select a batch of healthy gears and faulty gears as samples. After the gearbox is driven by a motor to run smoothly, the vibration signal and angular velocity signal of each gear are obtained through the vibration signal sensor and gear speed sensor arranged in step S1;

[0011] Classify and label the obtained gear vibration signals and angular velocity signals to obtain a labeled measured data set;

[0012] S3: Establish a gear dynamics model, solve the gear dynamics model through a numerical solution method, and obtain a labeled simulation data set;

[0013] S4: using the labeled measured data set and the labeled simulation data set to train the physical information neural network model, and obtain the trained physical information neural network model;

[0014] S5: import the labeled measured data set obtained in S2 and the labeled simulation data set obtained in S3 into the trained physical information neural network model to obtain a time-varying mesh stiffness change curve data set, classify and label the obtained time-varying mesh stiffness change curve data set, and obtain a labeled time-varying mesh stiffness curve data set;

[0015] S6: Calculate various features of the labeled time-varying mesh stiffness curve data set obtained in S5 as a fault classification training set;

[0016] S7: train a multi-classification algorithm model using the fault classification training set obtained in S6 to obtain a trained multi-classification algorithm model;

[0017] S8: Collecting gear vibration signals under the actual working state of the gearbox to obtain a gear vibration data set to be tested;

[0018] S9: Input the actual collected vibration data of the gear to be tested into the physical information neural network model trained in S4, output the time-varying meshing stiffness change curve, and then calculate the various features in step S6 by using the output time-varying meshing stiffness change curve. Finally, input the features into the multi-classification algorithm model trained in S7 to obtain the classification result, which is the gearbox fault result.

[0019] Further preferably, a gearbox fault diagnosis method based on physical information neural network comprises the following steps:

[0020] Step S1: Arrange vibration signal sensors around the gearbox bearings, and arrange gear speed sensors on the top side of the gears to obtain the vibration signals and angular velocity signals of the gearbox. Specifically, arrange vibration sensors under the bearing seats of the driving wheel and the driven wheel of the gearbox to collect vibration signals in the vertical and horizontal directions, and obtain the angular velocity signals of the gears through the gear speed sensors.

[0021] Step S2: Select a batch of healthy gears and faulty gears as samples, where the fault types of the faulty gears include tooth root fracture and tooth surface peeling, and obtain a labeled actual training data set. The specific operation is to use the faulty gearbox and the healthy gearbox to run for 10 seconds respectively, collect the corresponding gear vibration signal and angular velocity signal, and form a labeled measured data set.

[0022] Step S3: Establish a healthy gear dynamics model and various faulty gear dynamics models, calculate the theoretical gearbox vibration signal and angular acceleration signal through a numerical solution method, form a labeled simulation data set, and use the labeled simulation data set and the labeled measured data set to train a physical information neural network model with dynamic equations as physical constraints. The model uses a long short-term memory network (LSTM) as the neural network structure of the physical information neural network. The physical information neural network has 6 inputs (vibration signals and angular accelerations of the driving wheel and the driven wheel in the horizontal and vertical directions) and 1 output (time-varying meshing stiffness), and a trained physical information neural network model is obtained.

[0023] Step S4: inputting the labeled measured data set and the labeled simulation data set into the trained physical information neural network model to obtain a time-varying meshing stiffness change curve data set.

[0024] Step S5: label the fault classification of the time-varying mesh stiffness change data set curve to form a labeled time-varying mesh stiffness curve data set.

[0025] Step S6: extracting time domain features, frequency domain features and time-frequency mixed features from the labeled time-varying mesh stiffness curve data set obtained in step S5 as feature vectors for fault classification, and constructing a fault classification training set.

[0026] Step S7: Use the above fault classification training set to train a multi-classification algorithm model for the classification of gear faults.

[0027] Step S8: collecting the gear vibration signal of the gear box under the actual working state to obtain a gear vibration data set to be tested;

[0028] Step S9: Input the actual collected vibration data of the gear to be tested into the physical information neural network trained in S3, output the time-varying mesh stiffness change curve, and then calculate the various features in step S6 by using the output time-varying mesh stiffness change curve. Finally, input it into the multi-classification algorithm model trained in S7 to obtain the classification result, which is the gearbox fault result.

[0029] Improved features:

[0030] In step S1, the signals include: vibration signals in the horizontal and vertical directions below the gear driving wheel and driven wheel shaft, and gear angular velocity signals.

[0031] A high-frequency dynamic acquisition system was installed, and charge-type vibration sensors were used to collect vibration signals in the horizontal and vertical directions under the shafts of the driving and driven wheels of the gearbox. The signals were amplified using a signal amplifier and converted to a 10kHz acquisition device for data acquisition, which was finally saved as a healthy data set in TDM format.

[0032] In step S2, the faults include: tooth root fracture and tooth surface peeling.

[0033] The fault types include tooth root fracture and tooth surface spalling. After the fault is injected, the machine is kept rotating at a constant speed and running for 10 seconds. The vibration signal is collected to form a labeled measured data set.

[0034] The method for obtaining a gear with tooth root fracture fault is to partially cut the root of the gear using a grinding wheel.

[0035] The method for obtaining gears with tooth surface spalling failure is to use acidic solution for corrosion.

[0036] In step S3, when training the physical information neural network, two types of loss functions are used: a pure data-driven loss function and a physical-driven loss function based on the gear meshing dynamics equation to ensure the effectiveness of the physical constraints.

[0037] In the training optimization process of the physical information neural network in step S3, the loss function used is:

[0038] loss total =loss 1 +loss 2

[0039] Among them, loss 1 is a pure data-driven loss function, loss 2 A physically driven loss function built based on the gear mesh dynamics equations to better capture timing dependencies.

[0040] loss 1 and loss 2 The specific formula of the loss function is:

[0041]

[0042] loss 2 =r 1 +r 2 +r 3 +r 4 +r 5 +r 6

[0043]

[0044]

[0045]

[0046]

[0047]

[0048]

[0049] Among them, y, y pre is the actual result and the neural network prediction result, m 1 , m 2 is the mass of the driving wheel and the driven wheel, is the horizontal acceleration of the driving wheel and the driven wheel, is the vertical acceleration of the driving wheel and the driven wheel, c x1 , c x2 , c y1 , c y2 is the support damping of the driving wheel and the driven wheel in the x and y directions, is the horizontal speed of the driving wheel and the driven wheel, is the vertical speed of the driving wheel and the driven wheel, k x1 , k x2 , k y1 , k y2 is the support stiffness of the driving wheel and the driven wheel in the x and y directions, x 1 , x 2is the horizontal displacement of the driving wheel and the driven wheel, y 1 ,y 2 is the vertical displacement of the driving wheel and the driven wheel, is the angular acceleration of the driving wheel and the driven wheel, R 1 , R 2 is the base circle radius of the driving wheel and the driven wheel, J 1 , J 2 is the mass inertia moment of the driving wheel and the driven wheel, T 1 , T 2 is the torque of the driving wheel and the driven wheel, F m is the gear meshing force, and α is the gear meshing angle.

[0050] The time-varying meshing stiffness variation curve in step S6 has the following features extracted: time domain features, frequency domain features, and time-frequency mixed features.

[0051] Compared with the prior art, the present invention has the following advantages:

[0052] The present invention realizes high-precision classification of gearbox faults and can effectively identify common gear fault types by combining physical information neural networks with support vector machines. In particular, in the process of fault diagnosis, the diagnostic model has good robustness and accuracy by combining simulation data and measured data. By introducing physical information neural networks, the diagnostic model not only relies on data, but also combines the gear dynamics model, further improving the reliability and intelligence level of fault diagnosis. In addition, the neural network structure of the physical information neural network in the present invention adopts a long short-term memory neural network to replace the fully connected neural network structure in the traditional physical information neural network, which can better capture the timing dependency. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 Schematic diagram of the process of this method.

[0054] Figure 2 This is the gear meshing dynamics model used in this method.

[0055] Figure 3 Neuron structure diagram of LSTM used in this method.

[0056] Figure 4 For training and validation loss curves.

[0057] Figure 5 The diagram shows the time-varying mesh stiffness of gears under different predicted fault levels. DETAILED DESCRIPTION

[0058] In order to enable those skilled in the art to better understand the technical solutions in this specification, the technical solutions in the embodiments of this specification will be clearly and completely described below in conjunction with the drawings in the embodiments of this specification. Obviously, the described embodiments are only part of the embodiments of this application, not all of them.

[0059] In the description of the embodiments of the present application, words such as "for example" or "for example" are used to indicate examples, illustrations or explanations. Any embodiment or design described as "for example" or "for example" in the embodiments of the present application should not be interpreted as being more preferred or more advantageous than other embodiments or designs. Specifically, the use of words such as "for example" or "for example" is intended to present related concepts in a specific way.

[0060] In the description of the embodiments of the present application, the meaning of the term "multiple" refers to two or more. For example, multiple systems refer to two or more systems, and multiple screen terminals refer to two or more screen terminals. In addition, the terms "first" and "second" are used for descriptive purposes only and cannot be understood as indicating or implying relative importance or implicitly indicating the indicated technical features. Thus, the features defined as "first" and "second" may explicitly or implicitly include one or more of the features. The terms "include", "comprise", "have" and their variations all mean "including but not limited to", unless otherwise specifically emphasized.

[0061] like Figure 1 As shown, the present invention discloses a gearbox fault diagnosis method based on physical information neural network, which specifically includes the following steps:

[0062] S1: Arrange a vibration signal dynamic acquisition system to obtain the vibration signal and angular acceleration signal processed in the gearbox:

[0063] S11: Install a charge type vibration sensor below the fixed position of the driving wheel and driven wheel gear shaft of the gear box.

[0064] S12: Install the gear speed sensor 2mm above the driving wheel and driven wheel of the gearbox.

[0065] S13: Use a high-frequency dynamic acquisition system to respectively acquire acceleration signals in the horizontal and vertical directions below the gearbox driving wheel and driven wheel shafts and the angular velocity signals of the gears. The sampling frequency Fs used by the high-frequency dynamic acquisition system is 10kHz.

[0066] S2: Gear vibration signals and angular velocity signals are collected during the actual operation of the gearbox to form a measured data set:

[0067] S21: Select a batch of faulty gears and a batch of healthy gears and classify them.

[0068] S22: Use the servo motor to drive the faulty gear and the healthy gear in S21 to rotate at a uniform speed. The gear speed is 1200r / min. After running smoothly for 10 seconds, collect 5 seconds of gear vibration and angular velocity signals.

[0069] S23: The acquired data is timestamped according to the acquisition time and stored in a tdms format to form a measured data set including the health and fault status of the gear.

[0070] S3: Solve the kinetic equations to obtain the theoretical data training set:

[0071] S31: Establish Figure 2 The six-degree-of-freedom gear dynamics model shown, the differential equation used to solve is:

[0072]

[0073] Among them, m 1 , m 2 is the mass of the driving wheel and the driven wheel, is the horizontal acceleration of the driving wheel and the driven wheel, is the vertical acceleration of the driving wheel and the driven wheel, c x1 , c x2 , c y1 , c y2 is the support damping of the driving wheel and the driven wheel in the x and y directions, is the horizontal speed of the driving wheel and the driven wheel, is the vertical speed of the driving wheel and the driven wheel, k x1 , k x2 , k y1 , k y2 is the support stiffness of the driving wheel and the driven wheel in the x and y directions, x 1 , x 2 is the horizontal displacement of the driving wheel and the driven wheel, y 1 ,y 2 is the vertical displacement of the driving wheel and the driven wheel, is the angular acceleration of the driving wheel and the driven wheel, θ 1 ,θ 2 is the rotation angle of the driving wheel and the driven wheel, R 1 , R 2 is the base circle radius of the driving wheel and the driven wheel, J 1 , J 2 is the mass inertia moment of the driving wheel and the driven wheel, T 1 , T 2 is the torque of the driving wheel and the driven wheel, F mis the gear meshing force, and α is the gear meshing angle.

[0074] Some design parameters of the gear system are:

[0075] parameter Driving wheel Driven wheel Modulus / mm 27 73 Tooth width / mm 2 2 Pressure angle / (°) 20 20 <![CDATA[Mass inertia / (kg·m 2 )]]> 0.00017 0.0072 <![CDATA[Support stiffness / (N·m- 1 )]]> <![CDATA[1×10 7 ]]> <![CDATA[1×10 7 ]]> <![CDATA[Support damping / (N·m- 1 )]]> <![CDATA[1×10 3 ]]> <![CDATA[1×10 3 ]]>

[0076] S32: The dynamic model in S21 is numerically solved by the Runge-Kutta method, and the theoretical dynamic response is calculated.

[0077] S33: The dynamic response obtained by theoretical calculation and the time-varying meshing stiffness curve used for calculation are used as simulation data sets.

[0078] S4: Training physical information neural network model:

[0079] S41: Construct a physical information neural network model, which includes a long short-term memory network structure. The loss function of the physical information neural network includes the gear meshing dynamics model as a physical constraint. The specific loss function is as follows:

[0080] loss total =loss 1 +loss 2

[0081] Among them, loss 1 For a pure data-driven loss function, loss 2 A physically driven loss function built based on the gear mesh dynamics equations to better capture timing dependencies.

[0082] loss 1 and loss 2 The specific formula of the loss function is:

[0083]

[0084] loss 2 =r 1 +r 2 Ten r 3 +r 4 Ten r 5 +r 6

[0085]

[0086]

[0087]

[0088]

[0089]

[0090]

[0091] Among them, y, y pre is the actual result and the neural network prediction result, m 1 , m 2 is the mass of the driving wheel and the driven wheel, is the horizontal acceleration of the driving wheel and the driven wheel, is the vertical acceleration of the driving wheel and the driven wheel, c x1 , c x2 , c y1 , c y2 is the support damping of the driving wheel and the driven wheel in the x and y directions, is the horizontal speed of the driving wheel and the driven wheel, is the vertical speed of the driving wheel and the driven wheel, k x1 , k x2 , k y1 , k y2 is the support stiffness of the driving wheel and the driven wheel in the x and y directions, x 1 , x 2 is the horizontal displacement of the driving wheel and the driven wheel, y 1 ,y 2 is the vertical displacement of the driving wheel and the driven wheel, is the angular acceleration of the driving wheel and the driven wheel, R 1 , R 2 is the base circle radius of the driving wheel and the driven wheel, J 1 , J 2 is the mass inertia moment of the driving wheel and the driven wheel, T 1 , T 2 is the torque of the driving wheel and the driven wheel, F m is the gear meshing force, and α is the gear meshing angle.

[0092] In this example, a single-layer long short-term memory network (LSTM) is used as the structure of the neural network in the physical information neural network. The LSTM structure includes two layers of LSTM, where the first layer has 64 neurons and the second layer has 32 neurons. The LSTM neurons are as follows: Figure 3 As shown, a fully connected layer (Dense Layer) with 16 neurons follows, and finally outputs the time series signal of 1 neuron, and the input is 6-channel time series data with a length of 300.

[0093] S42: The design parameters and vibration response are used as the input of the model (300 length, 6 channels), and the time-varying mesh stiffness is used as the output of the model (300 length, 1 channel).

[0094] S43: Using the measured data set obtained in S23 and the simulated data set obtained in S23, the data set is divided into a training set, a test set, and a validation set at a ratio of 80%, 10%, and 10%, respectively. The model is trained using the training set, and the performance of the model is evaluated and optimized using the test set and the validation set. Finally, a trained physical information neural network model is obtained. The change of the loss function during the training process is shown in Figure 4 shown.

[0095] S5: Constructing a time-varying mesh stiffness curve dataset:

[0096] S51: Input the measured data set obtained in S15 and the simulation data set obtained in S23 into the trained physical information neural network model obtained in S33 to obtain the corresponding time-varying meshing stiffness curve to form a time-varying meshing stiffness change curve data set.

[0097] S52: Classify and label the time-varying mesh stiffness change curve data set obtained in S51 to obtain a labeled time-varying mesh stiffness curve data set.

[0098] S6: Construct a multi-classification support vector machine fault classification dataset:

[0099] S61: Extract features from the labeled time-varying mesh stiffness curve data set obtained in S52. A total of 32 features are selected. The extracted features are shown in the following table:

[0100]

[0101] In the table, N is the total number of signal points, and x(n) is the value corresponding to the nth sampling point of the signal.

[0102] S62: Classify and label the extracted data set to obtain a fault classification training set.

[0103] S7: Use the fault classification training set obtained in step S62 as training samples to train a multi-classification support vector machine to build a classification model from time-varying meshing stiffness characteristics to fault types.

[0104] S8: using the vibration signal dynamic acquisition system arranged in step S1 to collect the gear vibration signal under the actual working state of the gearbox, and obtain a gear vibration data set to be tested;

[0105] S9: Input the gear vibration data actually collected in step S8 into the physical information neural network model trained in step S4 to obtain Figure 5The predicted time-varying mesh stiffness curve of the gear under different fault degrees is shown. Then, the 32 features in step S6 are calculated for the output time-varying mesh stiffness change curve, and then these features are input into the support vector machine model trained in S7 to finally obtain the classification result, which is the gearbox fault result.

[0106] It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other specific forms without departing from the spirit or essential features of the invention. Therefore, the embodiments should be considered exemplary and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations falling within the meaning and scope of the equivalent elements of the claims be included in the invention. Any reference numeral in a claim should not be considered as limiting the claim to which it relates.

[0107] In addition, it should be understood that although the present specification is described according to implementation modes, not every implementation mode contains only one independent technical solution. This description of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment may also be appropriately combined to form other implementation modes that can be understood by those skilled in the art.

Claims

1. A gearbox fault diagnosis method based on physical information neural network, characterized in that: The following steps are involved: S1: Obtain the design parameters of the gears in the gearbox, arrange vibration signal sensors around the gearbox bearings, and arrange gear speed sensors on the top side of the gears; S2: Select a batch of healthy gears and faulty gears as samples. After the gearbox is driven by a motor to run smoothly, the vibration signal and angular velocity signal of each gear are obtained through the vibration signal sensor and gear speed sensor arranged in step S1; Classify and label the obtained gear vibration signals and angular velocity signals to obtain a labeled measured data set; S3: Establish a gear dynamics model, solve the gear dynamics model through a numerical solution method, and obtain a labeled simulation data set; S4: using the labeled measured data set and the labeled simulation data set to train the physical information neural network model, and obtain the trained physical information neural network model; S5: import the labeled measured data set obtained in S2 and the labeled simulation data set obtained in S3 into the trained physical information neural network model to obtain a time-varying mesh stiffness change curve data set, classify and label the obtained time-varying mesh stiffness change curve data set, and obtain a labeled time-varying mesh stiffness curve data set; S6: Calculate various features of the labeled time-varying mesh stiffness curve data set obtained in S5 as a fault classification training set; S7: train a multi-classification algorithm model using the fault classification training set obtained in S6 to obtain a trained multi-classification algorithm model; S8: Collecting gear vibration signals under the actual working state of the gearbox to obtain a gear vibration data set to be tested; S9: Input the actual collected vibration data of the gear to be tested into the physical information neural network model trained in S4, output the time-varying meshing stiffness change curve, and then calculate the various features in step S6 by using the output time-varying meshing stiffness change curve. Finally, input the features into the multi-classification algorithm model trained in S7 to obtain the classification result, which is the gearbox fault result.

2. The gearbox fault diagnosis method based on physical information neural network according to claim 1 is characterized in that: In step S2, the faults include: tooth root fracture and tooth surface peeling.

3. The gearbox fault diagnosis method based on physical information neural network according to claim 1 is characterized in that: In step S2, the gear vibration signal includes: vibration signals of the gear driving wheel and driven wheel shaft in the horizontal and vertical directions.

4. The gearbox fault diagnosis method based on physical information neural network according to claim 1 is characterized in that: In step S4, the physical information neural network model adopts a long short-term memory network as a network structure and a dynamic equation as a physical constraint, takes the gear vibration signal and angular velocity signal in the gearbox as input, and takes the time-varying meshing stiffness as output.

5. The gearbox fault diagnosis method based on physical information neural network according to claim 1 is characterized in that: In step S4, the physical information neural network model is trained, and the total loss function used is: loss total =loss1+loss2 Among them, loss1 is a pure data-driven loss function, and loss2 is a physical-driven loss function based on the gear meshing dynamics equation; The specific formulas of loss1 and loss2 loss functions are: loss2=r1+r2+r3+r4+r5+r6 Among them, y, y pre are the actual results and the predicted results of the neural network, m1 and m2 are the masses of the driving wheel and the driven wheel, is the horizontal acceleration of the driving wheel and the driven wheel, is the vertical acceleration of the driving wheel and the driven wheel, c x1 , c x2 , c y1 , c y2 is the support damping of the driving wheel and the driven wheel in the x and y directions, is the horizontal speed of the driving wheel and the driven wheel, is the vertical speed of the driving wheel and the driven wheel, k x1 , k x2 , k y1 , k y2 is the support stiffness of the driving wheel and the driven wheel in the x and y directions, x1, x2 are the horizontal displacements of the driving wheel and the driven wheel, y1, y2 are the vertical displacements of the driving wheel and the driven wheel, is the angular acceleration of the driving wheel and the driven wheel, R1, R2 are the base circle radii of the driving wheel and the driven wheel, J1, J2 is the mass inertia moment of the driving wheel and the driven wheel, T1, T2 are the torques of the driving wheel and the driven wheel, F m is the gear meshing force, and α is the gear meshing angle.

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